The Field Theory of Generalized Ferromagnet on the Hermitian Symmetric Spaces 1

We discuss the recent developments in the generalized continuous Heisenberg ferromagnet model formulated as a nonrelativistic field theory defined on the target space of the coadjoint orbits. Hermitian symmetric spaces are special because they provide completely integrable field theories in 1+1 dimension and self-dual Chern-Simons solitons and vortices in 2+1 dimension. Recently, an action principle of a nonrelativistic nonlinear sigma model with the target space of coadjoint orbits and its coupling with the Chern-Simons gauge field was proposed [1, 2]. The coadjoint orbits are naturally equipped with symplectic structure [3] and this can be used to construct the action for nonrelativistic field theories of generalized spins which are defined on them with arbitrary groups. The resulting models describe generalized Heisenberg ferromagnet in which the equation of motion satisfies the generalized Landau-Lifshitz (LL) equation. The Hermitian symmetric spaces [4] which are special types of the coadjoint orbits are especially interesting because they provide completely integrable field theories in 1+1 dimension [1] and self-dual Chern-Simons solitons and vortices in 2+1 dimension [2]. In this talk, I will present a review on the subject and discuss the related issues. This work was done in collaboration with Q-Han Park. We start with a brief summary of the phase space of the coadjoint orbits and the Hermitian symmetric space. Consider a cotangent bundle T G ∼= G × G∗ [5] of an arbitrary group G which can be regarded as the phase space for the generalized spin Talk given at 15th Symposium on Theoretical Physics: ”Field Theoretical Methods in Fundamental Physics”, Seoul, Korea, August 1996.